The new floricane-fruiting cultivar of blackberry called ‘A-1960’ is described herein. The new cultivar originated from a hand-pollinated cross of Ark. selections Ark.-1583×Ark.-1482 made in 1991. The seeds resulting from this controlled hybridization were germinated in a greenhouse in the spring of 1992 and planted in a field near Clarksville, Ark. (West-Central Arkansas). The seedlings fruited in the summer of 1994. One seedling, designated ‘A-1960’, was selected in 1994 for its very firm fruit with long shape, medium-large size, attractive fruit appearance, very good flavor, excellent plant health, and cane erectness.